Note that the complexity of working with continuous numbers was studied in various contexts.
To avoid confusion, let us make some remarks on the underlying machine model.
The underlying machine model for \(\exists \mathbb {R}\)  (over which sentences need to be decided and where reductions are performed) is the word RAM (or equivalently, a Turing machine) and not the real RAM  [1]} or the Blum-Shub-Smale model [2]}.
